Method and apparatus for transmitting and receiving a signal in MIMO broadcast channel with imperfect CSIT

ABSTRACT

A method and an apparatus for use in a multiuser radio communication system for transmitting and receiving a signal in a Multiple Input and Multiple Output (MIMO) broadcast channel with an imperfect Channel State Information at the Transmitter (CSIT) are provided. The signal transmission/reception method includes receiving CSI from each of a plurality of receivers, determining a transmit power and a transmission rate based on the CSI qualities of the plurality of receivers calculated from the CSI from each of the plurality of receivers, and transmitting the signal using the transmit power and the transmission rate. The present disclosure is capable of maximizing system throughput in wireless communication system.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims the benefit under 35 U.S.C. §119(e) of a U.S.Provisional application filed on Jan. 14, 2014 in the U.S. Patent andTrademark Office and assigned Ser. No. 61/927,193, the entire disclosureof which is hereby incorporated by reference.

JOINT RESEARCH AGREEMENT

The present disclosure was made by or on behalf of the below listedparties to a joint research agreement. The joint research agreement wasin effect on or before the date the present disclosure was made and thepresent disclosure was made as a result of activities undertaken withinthe scope of the joint research agreement. The parties to the jointresearch agreement are 1) SAMSUNG ELECTRONICS CO., LTD. and 2) IMPERIALINNOVATIONS LTD.

TECHNICAL FIELD

The present disclosure relates to a multiple antenna multiuser radiocommunication system. More particularly, the present disclosure relatesto transmission strategies for Multiple Input and Multiple Output (MIMO)broadcast channel with imperfect Channel State Information at theTransmitter (CSIT).

BACKGROUND

In Multi User-Multiple Input and Multiple Output (MU-MIMO) transmissions(for both Broadcast Channel and Interference Channel), the majorperformance drop is caused by the imperfect Channel State Information atthe Transmitter (CSIT). This is because in current standards the MU-MIMOtransmission strategy has been designed under the assumption of perfectCSI knowledge at the transmitter but is actually used in scenarios whereCSI is imperfectly known at the transmitter. Moreover, given the CSITfeedback mechanism in current standardizations, the accuracies of theCSIT vary across subbands and users depending on the availability ofwideband Precoding Matrix Indicator (PMI) and user-specific PMI. Aninteresting work is to design new transmission blocks that cope with theimperfect (instantaneous) CSIT and making use of the varying CSITqualities to benefit the performance.

The metric considered in this disclosure is the Degrees of Freedom(DoF). It can be interpreted as the number of interference-free streamstransmitted to each receiver. Mathematically, it is given by

${d = {\lim_{P->\infty}\frac{R}{\log_{2}P}}},$where R is the rate and P stands for the Signal-to-Noise Ratio (SNR).

To investigate the impact of the imperfect CSIT on the sum DoFperformance, the terminology, CSIT quality, is introduced. The CSITquality is considered within the range of 0 to 1, representing unknownCSIT and perfect CSIT respectively. Moreover, the CSIT qualities arelikely to vary across subbands and users. This setup can be interpretedas a practical deployment in line with Long Term Evolution (LTE) byconnecting the CSIT quality with the availability of wideband PMI andsubband PMI.

With the classical MU-MIMO transmission, the sum DoF performance isN₂−N₁+N₁ā+N₁ b, where ā and b respectively stand for the average CSITquality of receiver 1 and receiver 2 across all the subbands. If SingleUser-Multiple Input and Multiple Output (SU-MIMO) is performed, themaximal sum DoF performance will be N₂ if the transmitter only sendsmessages intended for receiver 2. The key ingredient of theachievability relies on the interference cancellation using the pastCSIT and channel output.

Moreover, the optimal strategy for the Multiple Input and Single Output(MISO) case despite the existence of imperfect CSIT, but DoF loss isincurred if it is reused in the MIMO case. The transmitted signal ismade up of private messages and common messages. Intuitively, since tworeceivers have different antennas, the number of common messages shouldbe limited by N₁, otherwise, Rx1 is unable to decode them. This causesspace resource wasted at Rx2 as it should have decoded N₂ streams ofcommon messages at a time.

The above information is presented as background information only toassist with an understanding of the present disclosure. No determinationhas been made, and no assertion is made, as to whether any of the abovemight be applicable as prior art with regard to the present disclosure.

SUMMARY

Aspects of the present disclosure are to address at least theabove-mentioned problems and/or disadvantages and to provide at leastthe advantages described below. Accordingly, an aspect of the presentdisclosure is to provide a signal transmission method and apparatusembodied in such a way that a User Equipment (UE) generates feedbackinformation in consideration of stochastic information on channelestimation error for use by a Base Station (BS) in a multi-antennamultiuser radio communication system.

Another aspect of the present disclosure is to provide a method andapparatus for feedback of channel information based on a channelestimation error prediction of a UE in a multi-antenna multiuser radiocommunication system.

Yet another aspect of the present disclosure is to provide a stochasticprecoding design and stochastic user selection method and apparatus of aBS based on a feedback channel information received from a user in amulti-antenna multiuser radio communication system.

In accordance with an aspect of the present disclosure, a signaltransmission/reception method of a terminal for use in a mobilecommunication system is provided. The signal transmission/receptionmethod includes receiving a reference signal transmitted by a basestation, estimating channel information based on the reference signal,predicting channel estimation error based on the channel information,and transmitting feedback information generated based on the channelestimation error to the base station.

In accordance with another aspect of the present disclosure, a signaltransmission/reception method of a base station for use in a mobilecommunication system is provided. The signal transmission/receptionmethod includes transmitting a reference signal to a terminal andreceiving feedback information generated based on the reference signal,wherein the feedback information is generated based on channelestimation error by the terminal on the basis of the reference signal.

In accordance with another aspect of the present disclosure, a terminalof transmitting receiving signals for use in a mobile communicationsystem is provided. The terminal includes a transceiver which transmitsand receives signals to and from a base station and a control unit whichcontrols the transceiver to receive a reference signal from the basestation, estimates channel information based on the reference signal,predicts channel estimation error based on the channel information, andcontrols the transceiver to transmit feedback information generatedbased on the channel estimation error.

In accordance with another aspect of the present disclosure, a BS oftransmitting receiving signals for use in a mobile communication systemis provided. The BS includes a transceiver which transmits and receivessignals to and from a terminal and a control unit which controls thetransceiver to transmit a reference signal to the terminal and receivefeedback information generated based on the reference signal, whereinthe feedback information is generated based on channel estimation errorby the terminal on the basis of the reference signal.

Other aspects, advantages, and salient features of the disclosure willbecome apparent to those skilled in the art from the following detaileddescription, which, taken in conjunction with the annexed drawings,discloses various embodiments of the present disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features, and advantages of certainembodiments of the present disclosure will be more apparent from thefollowing description taken in conjunction with the accompanyingdrawings, in which;

FIG. 1 is a diagram illustrating a system Model of (M, N₁, N₂) MultipleInput and Multiple Output (MIMO) Broadcast Channel with imperfectChannel State Information at the Transmitter (CSIT) according to anembodiment of the present disclosure;

FIG. 2 is a diagram illustrating a CSIT pattern according to anembodiment of the present disclosure;

FIG. 3 is a diagram illustrating received signals at each receiveraccording to an embodiment of the present disclosure; and

FIG. 4 is a diagram illustrating received signals at each receiveraccording to another embodiment of the present disclosure.

Throughout the drawings, it should be noted that like reference numbersare used to depict the same or similar elements, features, andstructures.

DETAILED DESCRIPTION

The following description with reference to the accompanying drawings isprovided to assist in a comprehensive understanding of variousembodiments of the present disclosure as defined by the claims and theirequivalents. It includes various specific details to assist in thatunderstanding but these are to be regarded as merely exemplary.Accordingly, those of ordinary skill in the art will recognize thatvarious changes and modifications of the various embodiments describedherein can be made without departing from the scope and spirit of thepresent disclosure. In addition, descriptions of well-known functionsand constructions may be omitted for clarity and conciseness.

The terms and words used in the following description and claims are notlimited to the bibliographical meanings, but, are merely used by theinventor to enable a clear and consistent understanding of the presentdisclosure. Accordingly, it should be apparent to those skilled in theart that the following description of various embodiments of the presentdisclosure is provided for illustration purpose only and not for thepurpose of limiting the present disclosure as defined by the appendedclaims and their equivalents.

It is to be understood that the singular forms “a,” “an,” and “the”include plural referents unless the context clearly dictates otherwise.Thus, for example, reference to “a component surface” includes referenceto one or more of such surfaces.

Some of elements are exaggerated, omitted or simplified in the drawingsand the elements may have sizes and/or shapes different from those shownin drawings, in practice. The same reference numbers are used throughoutthe drawings to refer to the same or like parts.

It will be understood that each block of the flowchart illustrationsand/or block diagrams, and combinations of blocks in the flowchartillustrations and/or block diagrams, can be implemented by computerprogram instructions. These computer program instructions may beprovided to a processor of a general purpose computer, special purposecomputer, or other programmable data processing apparatus to produce amachine, such that the instructions, which execute via the processor ofthe computer or other programmable data processing apparatus, createmeans for implementing the functions/acts specified in the flowchartand/or block diagram block or blocks. These computer programinstructions may also be stored in a non-transitory computer-readablerecording medium that can direct a computer or other programmable dataprocessing apparatus to function in a particular manner, such that theinstructions stored in the non-transitory computer-readable recordingmedium produce an article of manufacture including instruction meanswhich implement the function/act specified in the flowchart and/or blockdiagram block or blocks. The computer program instructions may also beloaded onto a computer or other programmable data processing apparatusto cause a series of operational steps to be performed on the computeror other programmable apparatus to produce a computer implementedprocess such that the instructions which execute on the computer orother programmable apparatus provide steps for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

Furthermore, the respective block diagrams may illustrate parts ofmodules, segments or codes including at least one or more executableinstructions for performing specific logic function(s). Moreover, itshould be noted that the functions of the blocks may be performed indifferent order in several modifications. For example, two successiveblocks may be performed substantially at the same time, or may beperformed in reverse order according to their functions.

The term “module” according to the various embodiments of thedisclosure, means, but is not limited to, a software or hardwarecomponent, such as a Field Programmable Gate Array (FPGA) or ApplicationSpecific Integrated Circuit (ASIC), which performs certain tasks. Amodule may advantageously be configured to reside on the addressablestorage medium and configured to be executed on one or more processors.Thus, a module may include, by way of example, components, such assoftware components, object-oriented software components, classcomponents and task components, processes, functions, attributes,procedures, subroutines, segments of program code, drivers, firmware,microcode, circuitry, data, databases, data structures, tables, arrays,and variables. The functionality provided for in the components andmodules may be combined into fewer components and modules or furtherseparated into additional components and modules. In addition, thecomponents and modules may be implemented such that they execute one ormore Central Processing Units (CPUs) in a device or a secure multimediacard.

The various embodiments of the present disclosure are to be regarded inan illustrative rather than a restrictive sense in order to helpunderstand the present disclosure. It is obvious to those skilled in theart that the present disclosure is applicable to other radiocommunication systems with appropriate modifications and changes withoutdeparting from the broader spirit and scope of the disclosure.

Although the description has been made with reference to particularembodiments, the present disclosure can be implemented with variousmodifications without departing from the scope of the presentdisclosure. Thus, the present disclosure is not limited to theparticular embodiments disclosed but will include the following claimsand their equivalents.

Although various embodiments of the disclosure have been described usingspecific terms, the specification and drawings are to be regarded in anillustrative rather than a restrictive sense in order to help understandthe present disclosure. It is obvious to those skilled in the art thatvarious modifications and changes can be made thereto without departingfrom the broader spirit and scope of the disclosure.

According to various embodiments of the present disclosure, the methodof at enhancing the sum Degrees of Freedom (DoF) performance based onthe integration of private messages and common messages as that in theMultiple Input and Single Output (MISO) case is suggested. The keyingredients lie in 1) always sending common messages and private symbolsintended for Rx2 with the DoF N₂ and try to send private symbols to Rx1with the DoF as large as possible; 2) design the transmission (e.g.,precoder, power allocation, etc.) of common messages in order to achievethe target DoF for common messages at both receivers.

The following notations are used in the rest of this text. Bold lowerletters stand for vectors whereas a symbol not in bold font represents ascalar. (•)^(T) and (•)^(H) represent the transpose and conjugatetranspose of a matrix or vector respectively. N(H) and R(H) respectivelystand for the null space and the range of H. E[•] refers to theexpectation of a random variable, vector or matrix. ∥•∥ is the norm of avector. f(P)˜P^(B) corresponds to

${\lim_{P->\infty}\frac{\log\;{f(P)}}{\log\; P}} = B$where P is SNR throughout the paper and logarithms are in base 2. (x)⁺means max(x, 0).gcd(N₁, N₂) is the greatest common devisor of integer N₁and N₂.

FIG. 1 is a diagram illustrating a system Model of (M, N₁, N₂) MultipleInput and Multiple Output (MIMO) Broadcast Channel with imperfectChannel State Information at the Transmitter (CSIT) according to anembodiment of the present disclosure.

2.1 System Model

Referring to FIG. 1, consider that a Tx has M antennas and tworeceivers, Rx1 and Rx2, which respectively have N₁ and N₂ antennas.Without loss of generality, assume N₁≦N₂. In any given subband j, thereceived signals at Rx1 and Rx2 are denoted by y_(j) and z_(j), wherey_(j) is a N₁×1 vector and z_(j) is a N₂×1 vector. ε_(j1) and ε_(j2) arerespectively the noise vector observed by Rx1 and Rx2. ε_(j1) and ε_(j2)are said to have a unit covariance matrix.

The transmitted signal in subband j is represented by a M×1 vector,s_(j), subject to the power constraint ∥s_(j)∥≦P, where P stands for theSNR throughout this document. Similar to the MISO case, the transmittedsignal consists of three kinds of messages:

Common messages I, denoted as c^(I) hereafter, are broadcast to bothusers and unique for each subband. They should be recovered by bothusers, but can be intended exclusively for user 1 or user 2;

Common messages II, denoted as c^(II) hereafter, should be recovered byboth users, but can be intended exclusively for user 1 or user 2. Unlikec^(I), c^(II) are broadcast twice, i.e., once in a subband where Rx1 candecode it and once in a subband where Rx2 is more capable of decodingit; and

Private message, denoted as u_(j) and v_(j) subband j, are respectivelyintended for user 1 or user 2 only. u_(j) and v_(j) are respectivelyN₁×1 and N₂×1 vectors. Private symbols intended for Rx1 and Rx2 arehereafter expressed as PS1 and PS2.

H_(j) and G_(j) respectively represent the channel of Rx1 and Rx2 insubband j. H_(j) is a M×N₁ complex Gaussian matrix with identitycovariance. H_(j) likely has full-rank. Similarly, G_(j) is a full-rankM×N₂ complex Gaussian matrix with identity covariance. As a start point,consider that M≧N₁+N₂.

Assume a general setup (valid for both Frequency Division Duplex (FDD)and Frequency Division Duplex (TDD)) where the transmitter obtains theCSI instantaneously, but with imperfectness, due to the estimation errorand/or finite rate in the feedback link.

Denoting Ĥ_(j) and Ĝ_(j) as the imperfect CSI of Rx1 and Rx2 in subbandj respectively, the CSI of user 1 and user 2 can be respectively modeledas:H _(j) =Ĥ _(j) +{tilde over (H)} _(j) ,G _(j) =Ĝ _(j) +{tilde over (G)}_(j)  Equation 1where {tilde over (H)}_(j) and {tilde over (G)}_(j) are thecorresponding error vectors. Each column of {tilde over (H)}_(j) has thesame normE[{tilde over (h)}_(ji) ^(H){tilde over (h)}_(ji)]=σ_(j1) ².Similarly, each column in {tilde over (G)}_(j) has the same normE[{tildeover (g)}_(ji) ^(H){tilde over (g)}_(ji)]=σ_(j2) ². Ĥ_(j) and Ĝ_(j) arerespectively independent of {tilde over (H)}_(j) and {tilde over(G)}_(j). The norm of the columns in Ĥ_(j) and Ĝ_(j) scale as P⁰ whenP→∞. Ĥ_(j) and Ĝ_(j) are obtained by both users and the transmitter, butH_(j) and G_(j) are only known by Rx1 and Rx2 respectively.

FIG. 2 is a diagram illustrating a CSIT pattern according to anembodiment of the present disclosure.

Referring to FIG. 2, to investigate the impact of the imperfect CSIT onthe DoF performance, assume that the variance of the norm of each columnin the error matrix exponentially scales with SNR, namely σ_(j1)²˜P^(−a) ^(j) and σ_(j2) ²˜P^(−b) ^(j) . a_(j) and b_(j) arerespectively interpreted as the qualities of the CSIT of Rx1 and Rx2 insubband j and given as follows:

$\begin{matrix}{{a_{j} = {- {\lim\limits_{P\rightarrow\infty}\frac{\log\;\sigma_{j\; 1}^{2}}{\log\; P}}}},{b_{j} = {- {\lim\limits_{P\rightarrow\infty}\frac{\log\;\sigma_{j\; 2}^{2}}{\log\; P}}}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

a_(j) and b_(j) vary within the range of [0,1]. a_(j)=1 (resp. b_(j)=1)is equivalent to perfect CSIT because the full DoF performance, i.e.,(d₁, d₂)=(N₁, N₂), can be achieved by simply performing Zero-ForcingBeamforming (ZFBF). a_(j)=0 (resp. b_(j)=0) is equivalent to no CSITbecause it means that the variance of the CSI error scales as P⁰, suchthat the imperfect CSIT cannot benefit the DoF when doing ZFBF. Besides,a_(j) and b_(j) vary across all the L subbands as shown in FIG. 2.

It is important to note the following quantities,E _(H,N(H) _(j) ₎ [H _(j) ^(H) N(H _(j))]=E _(H,N(H) _(j) ₎[(Ĥ _(j) ^(H)+{tilde over (H)} _(j) ^(H))N(H _(j))]=E _(H,N(H) _(j) ₎ [{tilde over(H)} _(j) ^(H) N(H _(j))]˜P ^(−a) ^(j) I _(N) ₁   Equation 3E _(G,N(G) _(j) ₎ [G _(j) ^(H) N(G _(j))]=E _(G,N(G) _(j) ₎[(Ĝ _(j) ^(H)+{tilde over (G)} _(j) ^(H))N(G _(j))]=E _(G,N(G) _(j) ₎ [{tilde over(G)} _(j) ^(H) N(G _(j))]˜P ^(−b) ^(j) I _(N) ₂   Equation 4as they are frequently used in the rest of this disclosure.2.2 Reusing MISO Case Scheme in the (M, N₁, N₂) MIMO Broadcast Channel

The main ingredient in the MISO case scheme is sending the privatesymbols via partial-ZFBF while transmitting common messages using theremaining power. The power and rate allocated to each private symbol andcommon symbol are determined based on the CSIT quality pattern. Besides,the common symbols are projected on the space spanned by the channels ofboth receivers. Generally, reusing the scheme in the (M, N₁, N₂) case,the transmitted signal in subband j is given by:

$\begin{matrix}{s_{j} = {\underset{\underset{{({P - P^{\max{({a_{j},b_{j}})}}})}I_{N_{1}}}{︸}}{{{CS}\; 1} + {{CS}\; 2}} + \underset{\underset{P^{b}j_{I_{N_{1}}}}{︸}}{{N\left( \hat{G} \right)}u_{j}} + \underset{\underset{P^{a}j_{I_{N_{2}}}}{︸}}{{N\left( \hat{H} \right)}v_{j}}}} & {{Equation}\mspace{14mu} 5}\end{matrix}$where N(Ĝ) is a M×N₁ precoding matrix spanning the null space of Ĝ.Since the nullity of Ĝ is M−N₂ and N₁<M−N₂ by assumption, it is possibleto send N₁ symbols to the orthogonal space of Ĝ. Similarly, N(Ĥ) is aM×N₂ precoding matrix spanning the null space of Ĥ. The received signalsare expressed as:

$\begin{matrix}{\mspace{79mu}{y_{j} = {\underset{\underset{{({P - P^{\max{({a_{j},b_{j}})}}})}I_{N_{1}}}{︸}}{H_{j}^{H}\left( {{{CS}\; 1} + {{CS}\; 2}} \right)} + \underset{\underset{P^{b}j_{I_{N_{1}}}}{︸}}{H_{j}^{H}{N\left( \hat{G} \right)}u_{j}} + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{{H_{j}^{H}{N\left( \hat{H} \right)}v_{j}} + \varepsilon_{j\; 1}}}}} & {{Equation}\mspace{14mu} 6} \\{z_{j} = {\underset{\underset{{({P - P^{\max{({a_{j},b_{j}})}}})}I_{N_{1}}}{︸}}{G_{j}^{H}\left( {{{CS}\; 1} + {{CS}\; 2}} \right)} + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{G_{j}^{H}{N\left( \hat{G} \right)}u_{j}} + \underset{\underset{P^{a}j_{I_{N_{2}}}}{︸}}{G_{j}^{H}{N\left( \hat{H} \right)}v_{j}} + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{\varepsilon_{j\; 2}}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

Counting all the private symbols sent in the L subbands, u_(1:L) achievethe

${{{DoF}\frac{\sum\limits_{j = 1}^{L}\;{N_{1}b_{j}}}{L}} = {N_{1}\overset{\_}{b}}},$while v_(1:L) achieve the

${{{DoF}\frac{\sum\limits_{j = 1}^{L}\;{N_{2}a_{j}}}{L}} = {N_{2}\overset{\_}{a}}},$because the private symbols are allocated with the power subject to thequality of the CSIT of their unintended receivers and are drowned by thenoise.

By treating the private symbols as noise, Rx1 can decode common messages(including both CS1 and CS2) with the

${{DoF}\frac{{N_{1}{\sum\limits_{j = 1}^{L}\; 1}} - b_{j}}{L}},$while Rx2 can decode them with the

${{DoF}\frac{{N_{2}{\sum\limits_{j = 1}^{L}\; 1}} - a_{j}}{L}},$As the common messages should be decodable by both receivers, theachievable DoF is N₁(1−max(ā, b)).

However, there are three problems limiting the sum DoF performance.

Space Resource Wasted at Rx2

With the MISO scheme, N₁ streams of common messages are sent at a timeso as to make them decodable at both receivers. This causes N₂−N₁dimensions of the received signals at Rx2 unused when decoding thecommon messages. Intuitively, the sum DoF performance can be improved ifthe resource (space, frequency and power) at Rx2 are fully employed tosince Rx2 has a larger inherent DoF (see Section 2.3 and MC 1 fordetails).

Achievable DoF of Common Messages

Definition of Inherent DoF for common messages: By treating the privatesymbols as noise, the inherent DoF for common messages at Rx1 and Rx2are respectively N₁(1−b) and N₂(1−ā).

The ideal DoF of the common messages is determined by the minimum valueof them, namely min(N₁(1−b), N₂(1−ā)). However, this is not achievableby reusing the MISO case scheme. Specifically, simply considering aone-subband case with N₁(1−b)=N₂(1−ā), Rx1 can decode each stream with ahigh DoF but the number of streams is small, while Rx2 can decode moresymbols but each with a small DoF. In the MISO scheme, the commonmessages are projected to the space spanning the channels of bothreceivers, therefore the common messages can only be decoded with theDoF min(N₁, N₂)×min(1−a, 1−b), namely min(N₁, N₂) streams are sent, eachof them has the DoF min(1−a, 1−b). Otherwise, at least one receiver willfail to decode them. Hence, a challenge is to design a transmissionstrategy to enhance the DoF of common messages such that both receiverscan decode them at the same time.

Transmission Strategy when Rx2 has a Larger Inherent DoF for the CommonMessages than Rx1

As a reminder, the MISO scheme said that when ā<b (or 1−ā>1−b, namelyRx2 has larger inherent DoF for the common messages than Rx1 if (N₁,N₂)=(1,1)) is replaced, the transmission strategy design starts withfinding b′_(1:L) such that b′=ā and b′_(j)≦b_(j), ∀j. This mechanism isinterpreted as decreasing the power and DoF allocated to PS1 and in turnthe DoF of the common messages is increased. This mechanism does notharm the sum DoF performance since both receivers are having the samedimensions in their received signal and the common messages contributeto the sum DoF equivalently as the private symbols.

However, this equivalence does not hold generally in the MIMO case dueto the discrepancy between the dimensions of the received signals ateach receiver. Considering one-subband scenario and Rx2 has a largerinherent DoF for the common messages, namely N₂(1−a)>N₁(1−b), the powerand DoF of PS2 is increased rather than doing the above mechanism.Although a DoF loss might be incurred for the private symbols intendedfor Rx1, the sum DoF improves because the private symbols intended forRx2 contribute more than Rx1.

Moreover, in multiple-subband scenario with varying CSIT qualities,increasing the DoF of PS2 in different subbands might cause a differentDoF loss of PS1. Consequently, the CSIT quality pattern will have animpact on the sum DoF performance, when Rx2 has a larger inherent DoFfor the common messages, namely N₂(1−ā)>N₁(1−b).

In the next section, the main claims are provided which address theproposed problems and lead to an enhanced sum DoF performance.

2.3 Main Claims (MC)

In this section, MC 1 addresses the first and third problems in Section2.2 while MC 2 and MC 3 address the solution to the second problem.

MC 1

A transmission strategy for two-receiver MIMO BC with imperfect CSITcomprises: 1) sending common messages (CS1 and CS2) and private messages(PS1 and PS2); 2) sending CS1, CS2 and PS2 with the DoF (d_(c), d₂),such that d_(c)+d₂=N₂; 3) sending PS1 with the DoF as high as possiblewhile guaranteeing d_(c)+d₂=N₂; and 4) based on the CSIT qualitypattern, determining the number, power and precoder for each kind ofmessage.

i. When N₁(1−b)≧N₂(1−ā), similar to the MISO case, the Tx1) sends commonmessages with the DoF N₁(1−b); 2) sends PS1 with the DoF N₁ b and PS2with the DoF N₂−N₁(1−b)=N₂(1−ā′); 3) the power allocated to PS2 in eachsubband is P^(a′) ^(j) , where a′_(j)≦a_(j).

ii. When N₁(1−b)<N₂(1−ā), unlike the MISO case, the DoF of commonmessages, PS1 and PS2 depend on the specific CSIT pattern.

iii. In MC 1.iii, the Tx sends PS2 in each subband with power P^(a′)^(j) , a′_(j)≧a_(j), and they are determined by finding the equalityregarding the inherent DoF at each Rx, namely

${\frac{1}{L}N_{1}{\sum\limits_{j = 1}^{L}\;\left( {1 - {\max\left( {b_{j},{a_{j} - a_{j}}} \right)}} \right)}} = {\frac{1}{L}N_{2}{\sum\limits_{j = 1}^{L}\;{\left( {1 - a_{j}^{\prime}} \right).}}}$

iv. In MC 1.iv, the value of either side of equality is determined asthe DoF for common messages.

v. In MC 1.iv, the calculation of a′_(j) starts from the subband withlowest b_(j).

vi. Unlike the MISO case, the common messages are transmitted using thestrategy given in MC 2 and MC 3 because the two receivers employ adifferent number of antennas to decode them.

Intuition 1 of MC 1: Sending PS1 with power higher than P^(b) ^(j) isequivalent with sending them with power P^(b) ^(j) . Generally, thefollowing equalities regarding the inherent DoF at each receiver andachieved DoF tuple (d_(c), d₁, d₂) can be written as:N ₁ =d _(c) +d ₁ +I ₁  Equation 8N ₂ =d _(c) +d ₂ +I ₂  Equation 9where I₂ is the interference caused by PS1 because they are sent withpower higher than P^(b) ^(j) . Now the power of PS1 is reduced to P^(b)^(j) . Consequently, there is no interference caused at Rx2 and in turnthe DoF of PS2 is increased to d′₂=d₂+I₂. At the same time, the DoF ofPS1 is decreased to d′₁=(d₁−I₂)⁺. Besides, the DoF of common messagesremains. The resulted sum DoF isd_(c)+d′₁+d′₂=d_(c)+d₂+I₂+(d₁−I₂)⁺=d₁+d₂+d_(c)+(I₂−d₁)+≧d₁+d₂+d_(c).

Intuition 2: When Rx2 has a larger inherent DoF, increasing the powerallocated to PS2 with the DoF d′_(c)+d′₂=N₂ results in a better sum DoFperformance. Here, considering Rx2 has a larger inherent DoF, I₂ in theequation N₂=d_(c)+d₂+I₂ stands for the unused resource.

Now, the DoF of PS2 d′₂ is increased, where d′₂=d₂+I₂. Since PS2consists of N₂ streams, the increment in each stream is

$\frac{I_{2}}{N_{2}}.$Moreover, as the received signal at Rx1 has N₁(≦N₂) dimensions, thisincrement results it

$\frac{I_{2}}{N_{2}} \times N_{1}$interference at Rx1. Hence, if

${d_{1} \geq {\frac{I_{2}}{N_{2}}N_{1}}},$the DoF of PS1 becomes

$d_{1}^{\prime} = {d_{1} - {\frac{I_{2}}{N_{2}}N_{1}}}$and the DoF of common messages remains. The equalities are

$N_{1} = {{d_{c} + d_{1}^{\prime} + {\frac{I_{2}}{N_{2}}N_{1}} + {I_{1}\mspace{14mu}{and}\mspace{14mu} N_{2}}} = {d_{c} + {d_{2}^{\prime}.}}}$Apparently, the resulted sum DoF is improved because the increase in theDoF of PS1 is greater than the loss of the DoF of PS2.

If

${d_{1} < {\frac{I_{2}}{N_{2}}N_{1}}},$the Tx sends PS2 and common messages with d′_(c)+d′₂=N₂ and d′₂>d₂+I₂.In this case, the DoF of PS1 becomes zero. The resulted sum DoF isd′_(c)+d′₂=(d_(c)+d₂+I₂)+d₁−d₁=(d_(c)+d₁+d₂)+(I₂−d₁)>d_(c)+d₁+d₂.

Combining these two intuitions, the key ingredients lie in alwayssending common messages and PS2 with the DoF d_(c)+d₂=N₂. Refer toSection 2.4.3 and 2.5.4 for more details.

MC 2

The transmission of CS1 comprises: properly determining the number,power and precoder of CS1, such that Rx1 decodes them using N₁ antennasand at the same time Rx2 decodes them with N₂ antennas. Morespecifically,

i. t{circumflex over (N)}₁{circumflex over (N)}₂CS1 are sent, where

${t = {\gcd\left( {N_{1},N_{2}} \right)}},{{\hat{N}}_{1} = \frac{N_{1}}{t}},{{{\hat{N}}_{2} = \frac{N_{2}}{t}};}$

ii. Each CS1 achieves the

${{{DoF}\;\Delta} = \frac{d_{c}}{t{\hat{N}}_{1}{\hat{N}}_{2}}},$where d_(c) is the target DoF of CS1; and

iii. With n=1, 2, . . . , t{circumflex over (N)}₁{circumflex over (N)}₂denoting the index of the CS1,

For those

${\left\lceil \frac{n}{N_{1}} \right\rceil = \left\lceil \frac{n}{N_{2}} \right\rceil},$the precoder is a vector in the space spanned by the channels of bothreceivers. The power allocated to them is

$P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}};$and

For those

${\left\lceil \frac{n}{N_{1}} \right\rceil > \left\lceil \frac{n}{N_{2}} \right\rceil},$the precoder is a summation of two parts: 1) a vector in the spacespanned by the channels of both receivers and with power

$P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}};$2) a vector in the null space of the channel of Rx1 and with power

$P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}.$

As a consequence of the precoder design and power allocation, Rx1 willobserves the CS1 in {circumflex over (N)}₂ power levels and each levelcontains N₁ unique CS1, while Rx2 will observes the CS1 in {circumflexover (N)}₁ power levels and each level contains N₂ unique CS1. Refer toSection 2.4.1 for details of the transmission design.

MC 3

Transmission of CS2 comprises: properly determining the number and powerof the CS2, such that Rx1 decodes them using N₁ antennas in the subbandwhere Rx1 has a large inherent DoF for the common messages, and Rx2decodes them using N₂ antennas in the subband where Rx2 has a largeinherent DoF for the common messages. More specifically,

i. t{circumflex over (N)}₁{circumflex over (N)}₂ common symbols aresent, where t=gcd(N₁, N₂),

${{\hat{N}}_{1} = \frac{N_{1}}{t}},{{{\hat{N}}_{2} = \frac{N_{2}}{t}};}$

ii. Each common symbol achieves the

${{{DoF}\;\delta} = \frac{d_{c}}{t\;{\hat{N}}_{1}{\hat{N}}_{2}}},$where d_(c) is the target DoF of CS2; and

iii. Without loss of generality, assume that Rx1 decodes them in subband1 while Rx2 recovers them in subband 2. These t{circumflex over(N)}₁{circumflex over (N)}₂ common symbols are sent twice:

In subband 1, the CS2 with index n (where n=1, 2, . . . , t{circumflexover (N)}₁{circumflex over (N)}₂) is sent with the precoder in the spacespanned by the channel of Rx1 and allocated with power

$P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}};$and

In subband 2, the CS2 with index n (where n=1, 2, . . . , t{circumflexover (N)}₁{circumflex over (N)}₂) is sent with the precoder in the spacespanned by the channel of Rx2 and allocated with power

$P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}.$

As a consequence, Rx1 will observe the CS2 in {circumflex over (N)}₂power levels and each level contains N₁ unique CS2 while Rx2 willobserve the CS2 in {circumflex over (N)}₁ power levels and each levelcontains N₂ unique CS2. In this way, Rx1 (resp. Rx2) can decode themfrom subband 1 (resp. subband 2) and remove the CS2 in subband 2 (resp.subband 1) so as to decode other symbols in subband 2 (resp. subband 1).Refer to Section 2.5.1 for more details.

Note that when N₁=N₂, MC 2 and MC 3 will become the same and similar asthat in MISO scheme.

2. 4 Transmission Strategy for One Subband Scenario

2.4.1. N₁(1−b)=N₂(1−a), Two Receivers have the Same Inherent DoF forCommon Messages

N₁ PS1 are sent with the power P^(b) and the precoder N(Ĝ), while N₂PS2are sent with power P^(a) and the precoder N(Ĥ).

According to MC 2, the t{circumflex over (N)}₁{circumflex over (N)}₂ CS1are received by Rx1 in N₂ power levels and each level contains N₁ uniqueCS1, while they are received by Rx2 in {circumflex over (N)}₁ powerlevels and each level contains N₂ unique CS1. Note that higher index ofthe level refers to a lower received power. The gap between two adjacentpower levels is Δ.

It is obvious that some CS1 are received in the same power level at eachreceiver (namely,

${\left\lceil \frac{n}{N_{1}} \right\rceil = \left\lceil \frac{n}{N_{2}} \right\rceil},$n is the index of the CS1) while others are received in the differencepower levels (namely,

$\left. {\left\lceil \frac{n}{N_{1}} \right\rceil > \left\lceil \frac{n}{N_{2}} \right\rceil} \right).$To this end, the precoder of these CS1 are designed to make use of thenull space of the CSIT of Rx1. More specifically,

{circumflex over (N)}₁{circumflex over (N)}₂t different CS1 are sent,each achieves the

${{{DoF}\;\Delta} = {\frac{1 - a}{{\hat{N}}_{1}} = \frac{1 - b}{{\hat{N}}_{2}}}};$

For those

${\left\lceil \frac{n}{N_{1}} \right\rceil = \left\lceil \frac{n}{N_{2}} \right\rceil},$the precoder is a vector in the space spanned by the channels of bothreceivers. The power allocated to them is

$P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}.$These CS1 can be expressed

$\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}};$

For those

${\left\lceil \frac{n}{N_{1}} \right\rceil > \left\lceil \frac{n}{N_{2}} \right\rceil},$the precoder is a summation of two parts: 1) a vector in the spacespanned by the channels of both receivers with power

$P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}};$2) a vector in the null space of Ĥ with power

$P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}.$These CS1 can be written as

${\left( {\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{R\left( {\hat{H},\hat{G}} \right)} + \underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{N\left( \hat{H} \right)}} \right)c_{n}},$where the first term is dominant at Rx1 while the second term isdominant at Rx2.

The transmitted signal is expressed as:

$\begin{matrix}{s = {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) + \left( {{\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + \underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{N\left( \hat{H} \right)}c_{n}}} \right) + \underset{\underset{P^{b}I_{N_{1}}}{︸}}{{N\left( \hat{G} \right)}u} + \underset{\underset{P^{a}I_{N_{2}}}{︸}}{{N\left( \hat{H} \right)}v}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

1) The received signal at Rx1 is:

$\begin{matrix}\begin{matrix}{y = {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) +}} \\{\left( {{\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + \underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta} - a}}{︸}}{H^{H}{N\left( \hat{H} \right)}c_{n}}} \right) +} \\{\underset{\underset{P^{b}I_{N_{1}}}{︸}}{H^{H}{N\left( \hat{G} \right)}u} + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{{H^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{1}}} \\{\approx {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) +}} \\{{\left( {{\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} +} \right)\underset{\underset{P^{b}I_{N_{1}}}{︸}}{H^{H}{N\left( \hat{G} \right)}u}} +} \\{\underset{\underset{P^{0}I_{N_{1}}}{︸}}{{H^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{1}}} \\{= {\left( {\sum\limits_{n = 1}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) + \underset{\underset{P^{b}I_{N_{1}}}{︸}}{H^{H}{N\left( \hat{G} \right)}u} + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{{H^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{1}}}} \\{= {\begin{pmatrix}{{\sum\limits_{n = 1}^{{\hat{N}}_{1}}\underset{\underset{P}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + {\sum\limits_{n = {N_{1} + 1}}^{N_{1} \times 2}\underset{\underset{P^{1 - \Delta}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + \ldots +} \\{\sum\limits_{n = {N_{1} + {({{\hat{N}}_{2} - 1})} + 1}}^{N_{1} \times {\hat{N}}_{2}}\underset{\underset{P^{1 - {{({{\hat{N}}_{2} - 1})}\Delta}}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}}\end{pmatrix} +}} \\{\underset{\underset{P^{b}I_{N_{1}}}{︸}}{H^{H}{N\left( \hat{G} \right)}u} + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{{H^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{1}}}\end{matrix} & {{Equation}\mspace{14mu} 11}\end{matrix}$

The approximation is due to the fact that H^(H) R(Ĥ, Ĝ)c_(n) is dominantcompared to H^(H)N(Ĥ)c_(n) at high SNR, namely

${1 - {\left( {\left\lceil \frac{n}{N_{1}} \right\rceil - 1} \right)\Delta}} \geq {1 - {\left( {\left\lceil \frac{n}{N_{2}} \right\rceil - 1} \right)\Delta} - {a \cdot {\forall{n \in {\left\lbrack {1,{{\hat{N}}_{1}{\hat{N}}_{2}t}} \right\rbrack.}}}}}$This can be verified by replacing

$\Delta = \frac{1 - b}{N_{2}}$and using the fact that

${{\left\lceil \frac{n}{N_{1}} \right\rceil - \left\lceil \frac{n}{N_{2}} \right\rceil} \leq {N_{2} - {N_{1}\mspace{14mu}{and}\mspace{14mu}{N_{1}\left( {1 - b} \right)}}}} = {{N_{2}\left( {1 - a} \right)}.}$

From the last equation, the N₁×N₂ CS1 received at Rx1 are written in{circumflex over (N)}₂ terms. Each term consists of N₁ different CS1with the same power. Specifically, the m th term (mε1, 2, . . .{circumflex over (N)}₂) contains the common symbols with index from(m−1)×N₁+1 to m×N₁, and their received power scale as P^(1−(M−1)Δ).

Decoding at Rx1 (MMSE-SIC): Since the dimension of received signal isN₁, Rx1 decode N₁ CS1 at a time by treating the r.h.s of them as noise.In this way, each CS1 recovered in the 1^(st) to the (N₂−1)th termachieves the DoF Δ. Each CS1 in the last term is recovered by treatingH^(H)N(Ĝ)u as noise and the DoF is 1−(N₂−1)Δ−b=Δ. After that, all theCS1 have been removed and the private symbols u are decoded only subjectto noise. Consequently, the DoF of the common message achieved at Rx1 is{circumflex over (N)}₁{circumflex over (N)}₂tΔ=N₁(1−b)=N₂(1−a) and theDoF of private symbols intended for Rx1 is N₁b.

2) The received signal at Rx2 is:

$\begin{matrix}\begin{matrix}{z = {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) +}} \\{\left( {{\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + \underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{G^{H}{N\left( \hat{H} \right)}c_{n}}} \right) +} \\{\underset{\underset{P^{0}I_{N_{1}}}{︸}}{G^{H}{N\left( \hat{G} \right)}u} + \underset{\underset{P^{a}I_{N_{2}}}{︸}}{G^{H}{N\left( \hat{H} \right)}v} + \underset{\underset{P^{0}I_{N_{2}}}{︸}}{\varepsilon_{2}}} \\{= {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) +}} \\{\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{G^{H}{N\left( \hat{H} \right)}c_{n}}} \right) + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{G^{H}{N\left( \hat{G} \right)}u} +} \\{\underset{\underset{P^{a}I_{N_{2}}}{︸}}{G^{H}{N\left( \hat{H} \right)}v} + \underset{\underset{P^{0}I_{N_{2}}}{︸}}{\varepsilon_{2}}} \\{\approx {\left( {{\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}{{n = 1},}}^{N_{2}}\underset{\underset{P}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + {\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}{{n = 1},}}^{N_{2}}\underset{\underset{P}{︸}}{G^{H}{N\left( \hat{H} \right)}c_{n}}}} \right) +}} \\{\left( {{\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}{{n = {N_{2} + 1}},}}^{N_{2} \times 2}\underset{\underset{P^{1 - \Delta}}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + {\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}{{n = {N_{2} + 1}},}}^{N_{2} \times 2}\underset{\underset{P^{1 - \Delta}}{︸}}{G^{H}{N\left( \hat{H} \right)}c_{n}}}} \right) + \ldots +} \\{\left( {{\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}{{n = {{N_{2}{({{\hat{N}}_{1} - 1})}} + 1}},}}^{N_{2} \times {\hat{N}}_{1}}\underset{\underset{P^{1 - {{({{\hat{N}}_{1} - 1})}\Delta}}}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + {\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}{{n = {{N_{2}{({{\hat{N}}_{1} - 1})}} + 1}},}}^{N_{2} \times {\hat{N}}_{1}}\underset{\underset{P^{1 - {{({{\hat{N}}_{1} - 1})}\Delta}}}{︸}}{G^{H}{N\left( \hat{H} \right)}c_{n}}}} \right) +} \\{\underset{\underset{P^{0}I_{N_{1}}}{︸}}{G^{H}{N\left( \hat{G} \right)}u} + \underset{\underset{P^{a}I_{N_{1}}}{︸}}{G^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{2}}\end{matrix} & {{Equation}\mspace{14mu} 12}\end{matrix}$

The approximation is due to the fact that G^(H)N(Ĥ)c_(n) is dominantcompared to G^(H)R (Ĥ, {tilde over (G)})c_(n) since

${1 - {\left( {\left\lceil \frac{n}{N_{1}} \right\rceil - 1} \right)\Delta}} \leq {1 - {\left( {\left\lceil \frac{n}{N_{2}} \right\rceil - 1} \right){\Delta.}}}$The CS1 received by Rx2 are written as the sum of N₁ brackets. N₂ uniqueCS1 are contained in each bracket and received with the same power.Specifically, the mth bracket (mε1, 2, . . . , {circumflex over (N)}₁)contains the common symbols with index from (m−1)×N₂+1 to m×N₂, andtheir received power scale as P^(1−(m−1)Δ).

Decoding at Rx2 is using MMSE-SIC and similar as that in Rx1.Consequently, the DoF of the common message achieved at Rx2 is{circumflex over (N)}₁{circumflex over (N)}₂tΔ=N₂(1−a)=N₁(1−b),identical to that decoded by Rx1. Besides, the DoF of PS2 is N₂a.Moreover, the sum DoF performance is N₂+N₁b.

Besides, it is important to note that a≧({circumflex over(N)}₂−{circumflex over (N)}₁)Δ (derived from

$\left. {{1 - {\left( {\left\lceil \frac{n}{N_{1}} \right\rceil - 1} \right)\Delta}} \geq {1 - {\left( {\left\lceil \frac{n}{N_{2}} \right\rceil - 1} \right)\Delta} - a}} \right)$is a necessary condition performing this transmission design. Otherwise,H^(H)R(Ĥ, Ĝ)c_(n) might not be dominant compared to H^(H)N(Ĥ)c_(n) atRx1.2.4.2. Rx1 has Larger Inherent DoF for the Common Messages Compared toRx2, Namely N₁(1−b)>N₂(1−a)

In this case, based on MC 1, the maximum achievable DoF of PS1 is N₁bwhile keeping d_(c)+d₂=N₂ because Rx1 has a larger inherent DoF. The sumDoF performance N₂+N₁ b has been shown as the optimal result since it isconsistent with the outer-bound.

The transmission strategy is constructed by calculating a′≦a such thatN₁(1−b)=N₂(1−a′). Then, using the scheme introduced in Section 2.4.1,the power allocated to the private symbols intended for Rx2 is P^(a′)and the DoF of each CS1 is determined as

$\Delta = {\frac{1 - a^{\prime}}{{\hat{N}}_{1}} = {\frac{1 - b}{{\hat{N}}_{2}}.}}$The transmitted signal is therefore written as:

$\begin{matrix}{s = {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) + \left( {{\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + \underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{N\left( \hat{H} \right)}c_{n}}} \right) + \underset{\underset{P^{b}I_{N_{1}}}{︸}}{{N\left( \hat{G} \right)}u} + \underset{\underset{P^{a^{\prime}}I_{N_{2}}}{︸}}{{N\left( \hat{H} \right)}v}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

Since N₁(1−b)≧N₂(1−a), it can be verified that a≧({circumflex over(N)}₂−{circumflex over (N)}₁)Δ and both receivers decode the CS1 withthe DoF N₁(1−b)=N₂(1−a′). Besides, N(Ĥ)v will not cause interference atRx1 as a′≦a. Consequently, the sum DoF is N₂(1−a′)+N₁b+N₂a′=N₂+N₁b.

2.4.3. Rx2 has a Larger Inherent DoF for the Common Messages, NamelyN₁(1−b)<N₂(1−a)

The scheme introduced in Section 2.4.1 does not work here since such b′satisfying 0≦b′≦b and N₁(1−b′)=N₂(1−a) does not necessarily exist.Therefore, based on MC 1, the power and DoF for PS2 is increased so asto decrease the inherent DoF for common messages at Rx2 till bothreceivers have same amount of inherent DoF for common messages. Althoughthis will cause interference at Rx1, it still benefits the sum DoFperformance as Rx2 has larger number of antennas because N₂≧N₁ and PS2contribute to the sum DoF performance more than PS1.

Once the power of PS2 is increased to P^(a′), the interference caused atRx1 is P^(a′−a). The inherent DoF for common messages is found byN₁(1−max(b, a′−a))=N₂(1−a′). Hence, the transmission strategy depends onwhether PS1 is drowned by the interference or not, namely therelationship between b and a′−a.

1) d_(c) is found when a′−a≦b, namely d_(c)=N₁(1−b)=N₂(1−a′) and thisleads to

$a \geq {\left( {N_{2} - N_{1}} \right) \times {\frac{1 - b}{N_{2}}.}}$

Compute a′>a, such that N₁(1−b)=N₂(1−a′);

Using the scheme introduced in Section 2.4.1, the power allocated to theprivate symbols intended for Rx2 is P^(a′) and the DoF of each CS1 isdetermined as

$\Delta = {\frac{1 - a^{\prime}}{{\hat{N}}_{1}} = {\frac{1 - b}{{\hat{N}}_{2}}.}}$The transmitted signal is therefore written as:

$\begin{matrix}{s = {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) + \left( {{\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + \underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{N\left( \hat{H} \right)}c_{n}}} \right) + \underset{\underset{P^{b}I_{N_{1}}}{︸}}{{N\left( \hat{G} \right)}u} + \underset{\underset{P^{a^{\prime}}I_{N_{2}}}{︸}}{{N\left( \hat{H} \right)}v}}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

Since

${a \geq {\left( {N_{2} - N_{1}} \right) \times \frac{1 - b}{N_{2}}}},$the necessary condition a≧({circumflex over (N)}₂−{circumflex over(N)}₁)Δ holds. The received signals are written as:

$\begin{matrix}{y = {\left( {\sum\limits_{m = 1}^{{\hat{N}}_{2}}{\sum\limits_{n = {{N_{1} \times {({m - 1})}} + 1}}^{N_{1} \times m}\underset{\underset{P^{1 - {{({m - 1})}\Delta}}}{︸}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}}} \right) + \underset{\underset{P^{b}I_{N_{1}}}{︸}}{H^{H}{N\left( \hat{G} \right)}u} + \underset{\underset{P^{a}\mspace{14mu} - a_{I_{N_{1}}}}{︸}}{H^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{1}}} & {{Equation}\mspace{14mu} 15} \\{z = {{\sum\limits_{m = 1}^{{\hat{N}}_{1}}\left( {{\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}{{n = {{N_{2}{({m - 1})}} + 1}},}}^{N_{2} \times m}\underset{\underset{P^{1 - {{({m - 1})}\Delta}}}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + {\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}{{n = {{N_{2}{({m - 1})}} + 1}},}}^{N_{2} \times m}\underset{\underset{P^{1 - {{({m - 1})}\Delta}}}{︸}}{G^{H}{N\left( \hat{H} \right)}c_{n}}}} \right)} + \underset{\underset{P^{0}I_{N_{1}}}{︸}}{G^{H}{N\left( \hat{G} \right)}u} + \underset{\underset{P^{a^{\prime}}I_{N_{2}}}{︸}}{G^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{2}}} & {{Equation}\mspace{14mu} 16}\end{matrix}$

As b≧a′−a, H^(H)N(Ĝ)u is the dominant part compared to H^(H)N(Ĥ)v.Hence, Both receivers can decode CS1 and the DoF achieved by CS1 isN₁(1−b)=N₂(1−a′). After removing c_(n), Rx1 decodes H^(H)N(Ĝ)u bytreating H^(H)N(Ĥ)v as noise and H^(H)N(Ĝ)u achieves the DoF N₁(b−a′+a).Rx2 recovers G^(H)N(Ĥ)v only subject to noise and the DoF of G^(H)N(Ĥ)vis N₂a′. Consequently, the sum DoF performance is

${N_{2} + {N_{1}\left( {b - a^{\prime} + a} \right)}} = {N_{2} + {{N_{1}\left( {a + {\frac{N_{2} - N_{1}}{N_{2}}\left( {b - 1} \right)}} \right)}.}}$

2) d_(c) is found when a′−a>b, namely d_(c)=N₁(1−a′+a)=N₂(1−a′) and thisleads to

$a < {\left( {N_{2} - N_{1}} \right) \times \frac{1 - b}{N_{2}}\text{:}}$

Compute a′>a, such that N₁(1−a′+a)=N₂(1−a′);

Using the scheme introduced in Section 2.4.1, the power allocated to PS2is P^(a′) and the DoF of each CS1 is determined as

${\Delta = {\frac{1 - a^{\prime}}{{\hat{N}}_{1}} = \frac{1 - a^{\prime} + a}{{\hat{N}}_{2}}}},$while PS1 is sent because b<a′−a. The transmitted signal is thereforewritten as:

$\begin{matrix}{s = {\left( {\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} \right) + \left( {{\sum\limits_{{n = 1},{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}}^{{\hat{N}}_{1}{\hat{N}}_{2}t}\underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\Delta}}}{︸}}{{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + \underset{\underset{P^{1 - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\Delta}}}{︸}}{{N\left( \hat{H} \right)}c_{n}}} \right) + \underset{\underset{P^{a^{\prime}}I_{N_{2}}}{︸}}{{N\left( \hat{H} \right)}v}}} & {{Equation}\mspace{14mu} 17}\end{matrix}$

Simply replacing

${\Delta = \frac{1 - a^{\prime} + a}{{\hat{N}}_{2}}},$the condition a≧({circumflex over (N)}₂−{circumflex over (N)}₁)Δ can beverified. The received signals are written as:

$\begin{matrix}{y = {\left( {\sum\limits_{m = 1}^{{\hat{N}}_{2}}{\sum\limits_{n = {{N_{1} \times {({m - 1})}} + 1}}^{N_{1} \times m}\underset{P^{1 - {{({m - 1})}\Delta}}}{H^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}}} \right) + \underset{\underset{P^{a^{\prime} - a_{I_{N_{1}}}}}{︸}}{H^{H}{N\left( \hat{H} \right)}v} + \varepsilon_{1}}} & {{Equation}\mspace{14mu} 18} \\{z = {{\sum\limits_{m = 1}^{{\hat{N}}_{1}}\left( {{\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} = {\lceil\frac{n}{N_{2}}\rceil}}{{n = {{N_{2}{({m - 1})}} + 1}},}}^{N_{2} \times m}\underset{\underset{P^{1 - {{({m - 1})}\Delta}}}{︸}}{G^{H}{R\left( {\hat{H},\hat{G}} \right)}c_{n}}} + {\sum\limits_{\underset{{\lceil\frac{n}{N_{1}}\rceil} > {\lceil\frac{n}{N_{2}}\rceil}}{{n = {{N_{2}{({m - 1})}} + 1}},}}^{N_{2} \times m}\underset{\underset{P^{1 - {{({m - 1})}\Delta}}}{︸}}{G^{H}{N\left( \hat{H} \right)}c_{n}}}} \right)} + \underset{P^{a}\mspace{14mu} I_{N_{2}}}{\underset{︸}{G^{H}{N\left( \hat{H} \right)}v}} + \varepsilon_{2}}} & {{Equation}\mspace{14mu} 19}\end{matrix}$

The decoding is similar as above and using MMSE-SIC. As no PS1 is sent,the sum DoF performance is N₂. Note that the CSIT does not make acontribution to the sum DoF performance compared to the scenario withoutCSIT of either receiver (the sum DoF is N₂ as well, achieved by sendingprivate symbols to Rx2 only), but the Tx can make use of the CSIT tosend common messages with the DoF N₂(1−a′). Since the common messagescan be regarded as intended for Rx1, this case is more meaningful in thescenario where the two receivers have their particular QoS target or theTx has to consider fairness between the receivers.

2.5 Transmission Strategy for Multiple Subband Scenario

2.5.1. Two-Subband Scenario N₂(1−ā)=N₁(1−b) and the Transmission of CS2

Here, consider a two-subband scenario with

${N_{2}\left( {1 - \frac{a_{1} + a_{2}}{2}} \right)} = {N_{1}\left( {1 - \frac{b_{1} + b_{2}}{2}} \right)}$and the transmission strategy can be easily extended to multiple-subbandscenario. Besides, assume N₁(1−b ₁)>N₂(1−a₁) and N₁(1−b₂)<N₂(1−a₂),namely Rx1 has larger inherent DoF for common messages in subband 1,while Rx2 has larger inherent DoF for common messages in subband 2. Thecase N₁(1−b₁)=N₂(1−a₁) and N₁(1−b₂)=N₂(1−a₂) can be solved independentlyand separately using the method introduced in Section 2.3. The scenario

${N_{2}\left( {1 - \frac{a_{1} + a_{2}}{2}} \right)} \neq {N_{1}\left( {1 - \frac{b_{1} + b_{2}}{2}} \right)}$will be discussed later.

Based on MC 3, the key ingredient in this scheme is to send CS2 with theDoF N₂(1−a₂)−N₁(1−b₂) (or equivalently N₁(1−b₁)−N₂(1−a₁)), such that Rx2can decode them from subband 2 and Rx1 recovers them from subband 1.

Transmission Strategy:

1) Private symbols: In each subband, N₁ private symbols are sent to Rx1with the ZF-precoder N(Ĝ_(j)) and the power P^(b) ^(j) , while N₂private symbols are sent to Rx2 with the ZF-precoder N(Ĥ_(j)) and thepower P^(a) ^(j) .

2) CS1 in subband 1: The DoF is determined to be N₂(1−a₁). According toMC 2 and as discussed in Section 2.4.1, {circumflex over(N)}₁{circumflex over (N)}₂t CS1 are sent to both receivers. These CS1in subband 1 are denoted as c_(1:N) ₁ _(N) ₂ ^(I)(1). Each CS1 has the

${{DoF}\;\Delta_{1}} = {\frac{1 - a_{1}}{{\hat{N}}_{1}}.}$The necessary condition, a₁≧({circumflex over (N)}₂−{circumflex over(N)}₁)Δ₁, can be verified using the fact that N₁(1−b₁)>N₂(1−a₁).

3) CS1 in subband 2: The DoF is determined to be N₁(1−b₂). The methodintroduced in Section 2.4.1 is not applicable here because the necessarycondition a₂≧({circumflex over (N)}₂−{circumflex over (N)}₁)Δ₂

$\left( {{{where}\mspace{14mu}\Delta_{2}} = \frac{1 - b_{2}}{{\hat{N}}_{2}}} \right)$does not hold in general. Hence, the joint design of CS1 and CS2 isconsidered here.

A. a₂≧b₂: The transmission of CS1 in subband 2 is divided into two partsas shown in Table 1.

TABLE 1 Notation DoF Number DoF/symbol Power Precoder Part 1 c_(n)^(I,1) (2) N₁ (r − b₂) {circumflex over (N)}₁{circumflex over (N)}₂t$\begin{matrix}{\Delta_{2,1} = \frac{r - b_{2}}{{\hat{N}}_{2}}} \\{= \frac{r - a_{2}}{{\hat{N}}_{1}}}\end{matrix}\quad$ Derived based on MC 2 and Section 2.4.1 Part 2 c_(n)^(I,2) (2) N₁ (1 − r) {circumflex over (N)}₁{circumflex over (N)}₁t$\delta = \frac{1 - r}{{\hat{N}}_{1}}$$P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\delta}}$ R(Ĥ₂, Ĝ₂)

Table 2 denotes the generation of two parts of CS1 in subband 2 whena₂≧b₂ where

$r = \frac{{N_{2}a_{2}} - {N_{1}b_{2}}}{N_{2} - N_{1}}$and r is derived from N₁(r−b₂)=N₂(r−a₂). c_(n) ^(I,1)(2) are generatedusing the method given in MC 2 and Section 2.4.1. c_(n) ^(I,2)(2) aresent with power higher than c_(n) ^(I,1)(2). c_(n) ^(I,2)(2) will bereceived by both receivers in {circumflex over (N)}₁ power levels andeach level contains N₁ unique c_(n) ^(I,2)(2).

B. a₂<b₂: The CS1 in subband 2 are generated similar as the second partof CS1 in the case a₂≧b₂. {circumflex over (N)}₁{circumflex over (N)}₁tCS1 are transmitted and the total DoF of them is N₁(1−b₂). They aredenoted as c_(n) ^(I)(2). Both receivers employ N₁ dimensions of thereceived signal to decode them and observe them in {circumflex over(N)}₁ power levels. The precoder is R(Ĥ₂, Ĝ₂). Each of them has the

${{DoF}\;\delta_{1}} = \frac{1 - b_{2}}{{\hat{N}}_{1}}$and is allocates with the power

$P^{1 - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\delta_{1}}}.$

4) CS2 Transmission: The DoF is Determined to beN₁(1−b₁)−N₂(1−a₁)=N₂(1−a₂)−N₂(1−b₂).

A. a₂≧b₂: Rx1 decodes CS2 using N₁ dimensions of the received signal insubband 1 while Rx2 employs N₂ N₁ dimensions to decode them (because N₁dimensions have been used to decode c_(n) ^(I,2)(2)). Based on MC 3,{circumflex over (N)}₁({circumflex over (N)}₂−{circumflex over (N)}₁)tCS2 are sent and they are denoted as c_(n) ^(II), n=1, 2, . . . ,{circumflex over (N)}₁({circumflex over (N)}₂−{circumflex over (N)}₁)t.Each of them has the target

${{{DoF}\frac{{N_{2}\left( {1 - a_{2}} \right)} - {N_{1}\left( {1 - b_{2}} \right)}}{{{\hat{N}}_{1}\left( {{\hat{N}}_{2} - {\hat{N}}_{1}} \right)}t}} = {\frac{1 - r}{{\hat{N}}_{1}} = \delta}},$same as c_(n) ^(I,2)(2):

In subband 1, c_(n) ^(II) is allocated with power

$P^{1 - {{\hat{N}}_{2}\Delta_{1}} - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\delta}}$and the precoder is R(Ĥ₁).

In subband 2, c_(n) ^(II) is allocated with power

$P^{1 - {{({{\lceil\frac{n}{N_{2} - N_{1}}\rceil} - 1})}\delta}}$and the precoder is R(Ĝ₂).

B. a₂<b₂: the transmission of CS2 is divided into two parts as shown inTable 2.

TABLE 2 Notation DoF Number DoF/symbol Power Precoder Part 1 c_(n)^(II,1) (N₂ − N₁)(1 − b₂) {circumflex over (N)}₁ ({circumflex over (N)}₂− {circumflex over (N)}₁)t$\delta_{1} = \frac{1 - b_{2}}{{\hat{N}}_{1}}$ Derived based on MC 3Part 2 c_(n) ^(II,2) N₂ (b₂ − a₂) {circumflex over (N)}₁{circumflex over(N)}₂t $\delta_{2} = \frac{b_{2} - a_{2}}{{\hat{N}}_{1}}$

Table 3 denotes CS2 generation when a₂<b₂. Note that c_(n) ^(II,1) isdecoded by Rx1 in subband 1 using N₁ dimensions and decoded by Rx2 insubband 2 using N₂−N₁ dimensions of the received signal (because N₁dimensions have been used to decode c_(n) ^(I)(2)).

In subband 1, is allocated with power

$P^{1 - {{\hat{N}}_{2}\Delta_{1}} - {{({\lceil{\frac{n}{N_{1}} - 1}\rceil})}\delta_{1}}}.$They will be received by Rx1 in {circumflex over (N)}₂−{circumflex over(N)}₁ power levels and each level contains N₁ unique c_(n) ^(II,1);c_(n) ^(II,2) is allocated with power

$P^{1 - {{\hat{N}}_{2}\Delta_{1}} - {{({{\hat{N}}_{2} - {\hat{N}}_{1}})}\delta_{1}} - {{({{\lceil\frac{n}{N_{1}}\rceil} - 1})}\delta_{2}}}$and observed by Rx1 in {circumflex over (N)}₂ power levels. Both c_(n)^(II,1) and c_(n) ^(II,2) are transmitted with the precoder R(Ĥ₁).

In subband 2, c_(n) ^(II,1) is allocated with power

$P^{1 - {{({{\lceil\frac{n}{N_{2} - N_{1}}\rceil} - 1})}\delta_{1}}}.$They will be received by Rx2 in {circumflex over (N)}₁ power levels andeach level contains N₂−N₁ unique C_(n) ^(II,1); c_(n) ^(II,2) isallocated with power

$P^{1 - {{\hat{N}}_{1}\delta_{1}} - {{({{\lceil\frac{n}{N_{2}}\rceil} - 1})}\delta_{2}}}$and observed by Rx2 in {circumflex over (N)}₁ power levels. Both c_(n)^(II,1) and c_(n) ^(II,2) are transmitted with the precoder R(Ĝ₂).

With the proposed strategy, the received signals at each receiver areillustrated in FIGS. 3 and 4 for a₂≧b₂ and a₂<b₂ respectively.

FIG. 3 is a diagram illustrating received signals at each receiver whena₂≧b₂ according to an embodiment of the present disclosure. Theupper-left, upper-right, lower-left and lower-right blocks respectivelyrefer to y₁, z₁, y₂ and z₂.

Referring to FIG. 3,

a) The private symbols are received with the power subject to the CSITquality of their unintended receiver;

b) According to step 2, Rx1 receives c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₂ _(t) ^(I)(1) with N₁ dimensions and{circumflex over (N)}₂ power levels, while Rx2 receives with N₂dimensions and {circumflex over (N)}₁ power levels;

c) Following step 3.A, c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₂ _(t) ^(I,1)(2) are received by Rx1 in N₁dimensions and r−b₂ channel use (namely {circumflex over (N)}₂ powerlevels), while they are received by Rx2 in N₂ dimensions and r−a₂channel use (namely {circumflex over (N)}₁ power levels);

d) Based on step 3.A, c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₁ _(t) ^(I,2)(2) are received by both receiversin 1−r channel use ({circumflex over (N)}₁ power levels) but onlyspanning N₁ dimensions; and

e) As in step 4.A, c_(1:{circumflex over (N)}) ₁_(({circumflex over (N)}) ₂ _(−{circumflex over (N)}) ₁ _()t) ^(II) arereceived by Rx2 in 1−r channel use ({circumflex over (N)}₁ power levels)and spanning the remaining N₂−N₁ dimensions in subband 2. At Rx1,c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)}) ₂_(−{circumflex over (N)}) ₁ _()t) ^(II) are received in N₁ dimensionsand {circumflex over (N)}₂−{circumflex over (N)}₁ power levels insubband 1.

Decoding (MMSE-SIC):

Rx1 decodes c_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂_(t) ^(I)(1), c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)}) ₂_(−{circumflex over (N)}) ₁ _()t) ^(II) and u₁ from y₁ using SIC; Withthe knowledge of c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)})₂ _(−{circumflex over (N)}) ₁ _()t) ^(II), Rx1 recoversc_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₁ _(t) ^(I,2)(2),c_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂ _(t) ^(I,1)(2)and u₂ from y₂; and

Rx2 decodes c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)}) ₂_(−{circumflex over (N)}) ₁ _()t) ^(II) and c_(1:{circumflex over (N)})₁ _({circumflex over (N)}) ₁ _(t) ^(I,2)(2) from z₂ by treatingc_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂ _(t) ^(I,1)(2)and 2 _(i) as noise. After that, c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₂ _(t) ^(I,1)(2) and v₂ are decoded using SIC;With the knowledge of c_(1:{circumflex over (N)}) ₁_(({circumflex over (N)}) ₂ _(−{circumflex over (N)}) ₁ _()t) ^(II),c_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂ _(t) ^(I)(1)and v₁ are decoded.

FIG. 4 is a diagram illustrating received signals at each receiver whena₂<b₂ according to another embodiment of the present disclosure. Theupper-left, upper-right, lower-left and lower-right blocks respectivelyrefer to y₁, z₁, y₂ and z₂.

Referring to FIG. 4,

a) The private symbols and c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₂ _(t) ^(I)(1) are received similar as in FIG.3;

b) As in step 3.B, c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₁ _(t) ^(I)(2) are received by both receiversin 1−b₂ channel use ({circumflex over (N)}₁ power levels) but onlyspanning N₁ dimensions;

c) According to step 4.B, c_(1:{circumflex over (N)}) ₁_(({circumflex over (N)}) ₂ _(−{circumflex over (N)}) ₁ _()t) ^(II,1)are received by Rx2 in 1−b₂ channel use ({circumflex over (N)}₁ powerlevels) and spanning the remaining N₂−N₁ dimensions in subband 2. AtRx1, c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)}) ₂_(−{circumflex over (N)}) ₁ _()t) ^(II,1) is received in N₁ dimensionsand N₂−N₁ power level in subband 1; and

d) Also based on step 4.B, c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₂ _(t) ^(II,2) are received by Rx1 in subband 1with N₁ dimensions and {circumflex over (N)}₂ power levels, while theyare received by Rx2 in N₂ dimensions and {circumflex over (N)}₁ powerlevels in subband 2.

Decoding:

Rx1 decodes c_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂_(t) ^(I)(1), c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)}) ₂_(−{circumflex over (N)}) ₁ _()t) ^(II,1), c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₂ _(t) ^(II,2) and u₁ from y₁ using SIC; Withthe knowledge of c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)})₂ _(−{circumflex over (N)}) ₁ _()t) ^(II,1) andc_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂ _(t) ^(II,2),Rx1 recovers c_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₁_(t) ^(I)(2) and private symbols from y₂; and

Rx2 decodes and c_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)})₂ _(−{circumflex over (N)}) ₁ _()t) ^(II,1) andc_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₁ _(t) ^(I)(2)from z₂ by treating c_(1:{circumflex over (N)}) ₁_({circumflex over (N)}) ₂ _(t) ^(II,2) and v₂ as noise. After that,c_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂ _(t) ^(II,2)and v₂ are decoded using SIC; With the knowledge ofc_(1:{circumflex over (N)}) ₁ _(({circumflex over (N)}) ₂_(−{circumflex over (N)}) ₁ _()t) ^(II,1) andc_(1:{circumflex over (N)}) _(1{circumflex over (N)}) ₂ _(t) ^(II,),c_(1:{circumflex over (N)}) ₁ _({circumflex over (N)}) ₂ _(t) ^(I)(1)and v₁ are decoded from z₁.

2.5.2. Extension to Multiple-Subband Scenario with N₂(1−ā)=N₁(1−b)

The private symbols and CS1 are transmitted in each subband followingthe footsteps 1) to 3) in Section 2.5.1. The DoF achieved by PS1 and PS2are respectively N₁ b and N₂ā, while CS1 achieves the

${DoF}{\frac{\sum\limits_{j = 1}^{L}{\min\left( {{N_{1}\left( {1 - b_{j}} \right)},{N_{2}\left( {1 - a_{j}} \right)}} \right)}}{L}.}$The CS2 are generated to achieve the

${DoF}{\frac{{\frac{1}{2}{\sum\limits_{j = 1}^{L}{\max\left( {{N_{1}\left( {1 - b_{j}} \right)},{N_{2}\left( {1 - a_{j}} \right)}} \right)}}} - {\min\left( {{N_{1}\left( {1 - b_{j}} \right)},{N_{2}\left( {1 - a_{j}} \right)}} \right)}}{L}.}$Rx1 decodes them from the subbands with N₁(1−b_(j))>N₂(1−a_(j)) whileRx2 decodes them from the subbands N₁(1−b_(j))<N₂(1−a_(j)). To this end,the procedure introduced in MISO scheme can be reused.

Briefly, the transmitter randomly pairs the subbands withN₁(1−b_(j1))>N₂(1−a_(j1)) and those with N₁(1−b_(j2))<N₂(1−a_(j2)), thenCS2 are generated using step 4) in Section 2.5.1 and achieving the DoFmin(q_(j1) ⁺, q_(j2) ⁻), where q_(j1) ⁺=N₁(1−b_(j1))−N₂(1−a_(j1)) andq_(j2) ⁺=N₂(1−a_(j2))−N₁(1−b_(j2)). After that, update q_(j1) ⁺=q_(j1)⁺−min(q_(j1) ⁺, q_(j2) ⁻) and q_(j2) ⁻=q_(j2) ⁻−min(q_(j1) ⁺, q_(j2) ⁻).Then repeat the procedure till all of q_(j1) ⁺ and q_(j2) ⁻ are zero.

2.5.3. Scenario N₂(1−ā)<N₁(1−b) (Rx1 has a Larger Inherent DoF forCommon Messages)

Similar as the discussion in Section 2.4.2 and based on MC 1, thetransmission strategy is designed by 1) Finding a′_(j)≦a_(j), ∀j, suchthat N₂(1−ā′)=N₁(1−b); 2) Reusing the design introduced in Section 2.5.2and replacing a_(j) with a′_(j). The sum DoF performance is N₂+N₁ b,which has been shown consistent with the outer-bound.

2.5.4. Scenario N₂(1−ā)>N₁(1−b) (Rx2 has a Larger Inherent DoF forCommon Messages)

The transmission strategy in this scenario follows MC 1. Similar as thediscussion in Section 2.4.3, d_(c)+d₂=N₂ is guaranteed by increasing thepower and DoF of PS2 and the power allocated to PS1 is fixed to be P^(b)^(j) . At the same time, in order to send PS1 with a DoF as high aspossible, the increment of PS2 is started from the subband with thelowest b₁. In this way, the DoF loss of PS2 can be minimized while thegap between the inherent DoF for common messages at each receiver isreducing. Specifically, the footsteps are given as follows:

Ordering the subband by b_(j) in ascending order. π(j) is used to denotethe subband with the jth lowest b_(j);

i=1;

Increasing the power and DoF of PS1 in subband π(i) to a′_(π(i)), wherea_(π(i))<a′_(π(i))≦1,

If Σ_(j=1) ^(i−1) N₁ min(a_(π(j)), 1−b_(π(j)))+Σ_(j=i+1) ^(L)N₁(1−b_(π(j)))+N₁(1−max(b_(π(i)), a′_(π(i))−a_(π(i))))=Σ_(j=i+1) ^(L)N₂(1−a_(π(j)))+N₂(1−a′_(π(i))), go to step d and seta′_(π(j>i))=a_(π(j>i));

Else, set a′_(π(i))=1 and i=i+1.

Construct the transmission strategy with a′_(1:L) and b_(j) as discussedin Section 2.5.2.

The l.h.s and r.h.s of the equation in step c respectively stand for theinherent DoF for common messages at Rx1 and Rx2 after increasinga_(π(1:i)) to a′_(π(1:i)). Since increasing a_(i) to a′_(i) strictlyreduces the gap between inherent DoF at Rx2 and Rx1, a new equality forthe common messages will be found. Finally, the sum DoF performance isgiven by:

$N_{2} + {N_{1}{\frac{\sum\limits_{j = 1}^{L}\left( {b_{j} - a_{j}^{\prime} + a_{j}} \right)^{+}}{L}.}}$

Note that in the multiple subbands scenario with the CSIT qualitya_(i:L)=a and b_(i:L)=b across the subbands, the transmission strategyproposed in this section results in a better DoF performance compared tousing the single-subband transmission (see Section 2.4.3) individuallyin each subband. Especially for the case

${{a + {\frac{{N_{2} - N_{1}}\;}{N_{2}}\left( {b - 1} \right)}} < 0},$the sum DoF is N₂ in Section 2.4.3, while the multiple-subbandtransmission strictly outperforms this as the Tx can always transmit PS1in some subbands. Hence, unlike the MISO case, the specific CSIT patternnot only impact the transmission strategy, but also the sum DoFperformance.2.6 Signaling Mechanisms Needed to Operate the Transmission Strategies

CSIT Quality Feedback

As in Section 2.1, the CSIT quality in MIMO case is defined based on thenorm of each column in the channel matrix, namely the channel vectorfrom the transmission antenna array to each receive antenna. Thisdefinition is similar as that in MISO case and moreover assume thechannel estimation and quantization at the receive antenna arestatistically equivalent. Hence, the similar signaling mechanismsregarding the CSIT quality feedback in MISO case can be reused.

Specifically, depending on the PMI report mode, if wideband PMI (PUCCH1-1-1, PUCCH 1-1-2 and PUSCH 3-1) and/or all subband PMI (PUSCH 3-2 andPUSCH 1-2) are available, each receiver reports a single value of theCSI accuracy as it is assumed the same across all subbands; if one orseveral subband PMI is reported (but not all), one or several estimatesof the CSIT accuracies are reported.

Transmission Mode Indication

Once the Tx collects the CSI and their qualities from all the receivers,it has to perform user-selection and subband grouping, then makes adecision to the transmission strategy. The similar DCI format as theMISO case can be reused, regarding the size of the transmission block(as it depends on CSIT pattern of the co-scheduled users), the kinds ofmessages (CS1, CS2 and PS1, PS2), and the number of messages, themodulation and coding scheme of all CS and PS intended for the user,information about whether common message is intended for the user ornot, the transmit power of each message. In addition, in the decision ofthe transmission mode, the following issue should be taken into account:

The format of CS: Based on MC 2 and MC 3, the transmission of CSconsists of several power levels depending on the value of N₁ and N₂.Hence, to perform the decoding, each receiver should know: 1) the numberof power levels of the common messages they have observed (for instance,Rx1 should know N₂ and Rx2 should know {circumflex over (N)}₁ accordingto MC 2, MC 3 and Section 2.4.1) as this relates to the number of stagesin the SIC; 2) the received power at each level (for instance, the powerreceived in the mth level P^(1−(m−1)Δ) according to Section 2.4.1) asthis impacts the MMSE filter used in each stage of SIC.

While the present disclosure has been shown and described with referenceto various embodiments thereof, it will be understood by those skilledin the art that various changes in form and details may be made thereinwithout departing from the spirit and scope of the present disclosure asdefined by the appended claims and their equivalents.

What is claimed is:
 1. A method for transmitting a signal based onchannel state information (CSI) qualities of at least two of a pluralityof receivers, the method comprising: receiving CSI from each of the atleast two of the plurality of receivers including a first receiver and asecond receiver; determining a transmit power and a transmission ratebased on the CSI qualities of the at least two of the plurality ofreceivers calculated from the CSI from each of the at least two of theplurality of receivers; and transmitting the signal using the transmitpower and the transmission rate, wherein the signal comprises a userspecific message, a first common message, and a second common message,and wherein the second common message is transmitted to the at least twoof the plurality of receivers using the transmission rate determined bya difference between a maximum CSI quality of a receiver among the atleast two of the plurality of receivers and a CSI quality of eachreceiver to compensate for the difference between the maximum CSIquality of the receiver among the at least two of the plurality ofreceivers and the CSI quality of each receiver.
 2. The method of claim1, wherein the user specific message is transmitted to a specificreceiver using the transmit power and the transmission rate determinedby a CSI quality of the specific receiver.
 3. The method of claim 1,wherein the first common message is transmitted to the at least two ofthe plurality of receivers using the transmit power and the transmissionrate determined by a maximum CSI quality of a receiver among the atleast two of the plurality of receivers to compensate for a differencebetween a perfect CSI quality and the maximum CSI quality of thereceiver among the at least two of the plurality of receivers.
 4. Anon-transitory computer-readable recording medium having embodiedthereon a computer program that when executed by a computer causes thecomputer to perform the method of claim
 1. 5. A transmitter apparatusfor transmitting a signal based on channel state information (CSI)qualities of at least two of a plurality of receivers, the apparatuscomprising: a transceiver for transmitting and receiving signals to andfrom the at least two of the plurality of receivers; and a controllerconfigured to receive CSI from each of the at least two of the pluralityof receivers, to determine a transmit power and a transmission ratebased on the CSI qualities of the at least two of the plurality ofreceivers calculated from the CSI from each of the at least two of theplurality of receivers, and to transmit the signal using the transmitpower and the transmission rate, wherein the signal comprises a userspecific message, a first common message, and a second common message,and wherein the second common message is transmitted to the at least twoof the plurality of receivers using the transmission rate determined bya difference between a maximum CSI quality of a receiver among the atleast two of the plurality of receivers and a CSI quality of eachreceiver to compensate for the difference between the maximum CSIquality of the receiver among the at least two of the plurality ofreceivers and the CSI quality of each receiver.
 6. The apparatus ofclaim 5, wherein the controller is further configured for transmittingthe user specific message to a specific receiver using the transmitpower and the transmission rate determined by a CSI quality of thespecific receiver.
 7. The apparatus of claim 5, wherein the controlleris further configured to transmit the first common message to the atleast two of the plurality of receivers using the transmit power and thetransmission rate determined by a maximum CSI quality of a receiveramong the at least two of the plurality of receivers to compensate for adifference between a perfect CSI quality and the maximum CSI quality ofthe receiver among the at least two of the plurality of receivers.